On the sharp dimension estimate of CR holomorphic functions in Sasakian Manifolds

TL;DR

This paper investigates the maximum possible dimension of CR holomorphic functions on complete noncompact Sasakian manifolds with nonnegative pseudohermitian bisectional curvature, extending Yau's uniformization conjecture to CR geometry.

Contribution

It provides the first sharp dimension estimates for CR holomorphic functions in Sasakian manifolds with nonnegative pseudohermitian bisectional curvature, advancing the understanding of CR analogues of complex geometric results.

Findings

Established sharp dimension bounds for CR holomorphic functions.

Extended Yau's uniformization conjecture to CR geometry.

Provided new insights into the structure of Sasakian manifolds.

Abstract

This is the very first paper to focus on the CR analogue of Yau's uniformization conjecture in a complete noncompact pseudohermitian $(2n+1)$-manifold of vanishing torsion (i.e. Sasakian manifold) which is an odd dimensional counterpart of K\"{a}hler geometry. In this paper, we mainly deal with the problem of the sharp dimension estimate of CR holomorphic functions in a complete noncompact pseudohermitian manifold of vanishing torsion with nonnegative pseudohermitian bisectional curvature.